In Exercises 31–38, factor completely.

3x⁴ + 54x³ + 135x²

Question

In Exercises 31–38, factor completely.

3x⁴ + 54x³ + 135x²

Accepted Answer

Welcome back. I am so glad you're here. Were asked to perform factory ization. In the given expression, the expression were given is two X to the fourth plus 28 X cubed plus 26 X squared. And our answer choices are answer choice A two X squared times parentheses, X plus one end parentheses, times new parentheses, X minus 13 and parentheses. Answer choice B two X squared times parentheses, X minus one end parentheses, times new parentheses, X plus 13 and parentheses. Answer choice C two X squared times parentheses, X minus one end parentheses, times new parentheses, X minus 13 and parentheses and answer choice D two X squared times parentheses, X plus one and parentheses times new parentheses X plus 13. Alright. The first step in factoring is that we take a look at each of these three terms and ask ourselves if there's anything that they have in common that we could factor out. So let's start off looking at these coefficients. We have 2:28 and 26. And the greatest common factor between those three is to, we can divide all three of them by two, meaning we can factor out a two. Now let's take a look at the variables, we have X to the fourth X to the third and X squared. And when we are comparing variables, we are going to look for the one with the lowest degree, the lowest degree between 43 and two is two. So that means we are going to factor out or divide all three of them by X squared. Alright, that's as much as we can factor out. So now we want to see what's left over once we've done that division. So we can go ahead and start with the first term. We're taking two X to the fourth divided by two X squared. Two, divided by two is one and X to the fourth divided by X squared. Well, we recall from previous lessons when we are dividing variables with like bases. Those are the X is we're going to subtract their exponents. So four minus two is two. We're left with X squared. That's a one X squared. We don't usually put that one there. So we can just leave it as X squared right now. The second term we have 28 X cubed divided by two X squared, 28 divided by two is 14. So we have plus 14 and then X cubed divided by X, that's three minus this unwritten one here. So that should be X squared. So it's X cubed divided by X squared and three minus two is one. So we're left with X to the first, which is just X. So that's a 14 X left over from the second term. And now for the third time, we have 26 X squared divided by two X squared, 26 divided by two is 13. So we have plus 13, an X squared divided by X squared. That's two minus two, which is zero. We have X to the zero and anything raised to the zero power is one. So we have 13 times one, but we don't need to put the times one because that's just 13. And next step in factoring is, will bring down this two X square that's in front. And we set up two sets of parentheses. And we are going to ask ourselves what times what gives us X squared. Those are our firsts and for X squared, that's going to be X times X now because there were no coefficients to that X squared, we get to move right on for our lasts and we ask ourselves what times what equals a positive 13. But when added together equals a positive 14, And that is going to be a plus one and a plus 13. And if we want, we can check this really quickly with foil. Our firsts are X times X that's X squared are outside our X times 13, that's plus 13 X. Our insides are one times X that's plus X and R lasts are one times 13 which is plus 13 adding together like terms in the middle here. So we'll bring down X squared, then we have plus 13 plus one, X plus 13, X plus one X is plus 14 X and then bring down the plus 13 and that does match, we factored it correctly. So our answer is two X squared times parentheses, X plus one end, parentheses times new parentheses, X plus 13 and parentheses which matches with answer choice D while done, we'll catch you on the next one.

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In Exercises 31–38, factor completely. 3x⁴ + 54x³ + 135x²